# Gaussian curvature

Gaussian curvature of a point on a surface is the product of the principal curvatures, k1 and k2 of the given point.

Symbolically, the Gaussian curvature K is defined as

[itex] K = k_1 k_2 [itex].

It is also given by

[itex]K = \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\rangle}{\det g}[itex]

where [itex]\nabla_i = \nabla_{{\mathbf e}_i}[itex] is the covariant derivative and g is the metric tensor.

At a point p on a regular surface in [itex]\mathbb{R}^3[itex], the Gaussian curvature is also given by

[itex]K(\mathbf{p}) = \det(S(\mathbf{p}))[itex]

where S is the shape operator.

• Art and Cultures
• Musical Instruments (http://academickids.com/encyclopedia/index.php/List_of_musical_instruments)
• Countries of the World (http://www.academickids.com/encyclopedia/index.php/Countries)
• Ancient Civilizations (http://www.academickids.com/encyclopedia/index.php/Ancient_Civilizations)
• Industrial Revolution (http://www.academickids.com/encyclopedia/index.php/Industrial_Revolution)
• Middle Ages (http://www.academickids.com/encyclopedia/index.php/Middle_Ages)
• United States (http://www.academickids.com/encyclopedia/index.php/United_States)
• World History (http://www.academickids.com/encyclopedia/index.php/History_of_the_world)
• Human Body (http://www.academickids.com/encyclopedia/index.php/Human_Body)
• Physical Science (http://www.academickids.com/encyclopedia/index.php/Physical_Science)
• Social Studies (http://www.academickids.com/encyclopedia/index.php/Social_Studies)
• Space and Astronomy
• Solar System (http://www.academickids.com/encyclopedia/index.php/Solar_System)